A Short Course on Quantum Mechanics and Methods of Quantization 35
Thus:
TrH[A]=
∫
dμ(z)A(z∗,z). (1.160)
This formula is very useful in some applications we will encounter in the following.
Example 1.3.1.
(i) The “delta”-operator:
δ(z∗−z∗ 0 ):ψ(z∗)→ψ(z∗ 0 ) (1.161)
can be written as:
ψ(z 0 ∗)=〈z 0 |ψ〉=〈z 0 |
(∫
dμ(z)|z〉〈z|
)
|ψ〉=
∫
dμ(z)〈z 0 |z〉ψ(z∗)
=
∫
dμ(z)ez
∗ 0 z
ψ(z∗), (1.162)
showing that its kernel is given by:ez
0 ∗z
.
(ii) The kernel of the annihilation operatorais given by:
〈z|a|z′〉=z′〈z|z′〉=z′ez
∗z′
. (1.163)
(iii) The kernel of the creation operatora†is given by:
〈z|a†|z′〉=z∗〈z|z′〉=z∗ez
∗z′
. (1.164)
(iv) The kernel of the number operatora†ais given by:
〈z|a†a|z′〉=z∗z′〈z|z′〉=z∗z′ez
∗z′
. (1.165)
(v) More generally, the kernel of any operator of the form^19 :
K=
∑
pq
kpq(a†)paq (1.166)
is simply given by the expression:
〈z|K|z〉=
∑
pq
kpq〈z|(a†)paq|z〉=
∑
pq
kpq(z∗)p(z′)qez
∗z′
. (1.167)
(^19) A polynomial ina, a†in which the creation operators are all on the left of the annihilation
ones is said to be in its normal form.